RATIONAL APPROXIMATIONS TO IRRATIONALS
ALEXANDER OPPENHEIM
It is well known that if p/q is a convergent to the irrational number
x, then \x—p/q\ < 1/q2. The immediate converse is of course false but
I have not seen in the literature 1 any statement of the converse which
is given below.
1. If p and q are coprime, q>0, and if \x —p/q\
then necessarily p/q is one of the three (irreducible) fractions
THEOREM
P'/q',
(p' + P")/(q' + q"),
<l/q2,
(P' ~ P")/(q' ~ ?").
where p"/q", p'/q' are two consecutive convergents to the irrational x.
One at least of the two f radions (p'+ep")/(q' + eq") where €= ± 1 satisfies the inequality.
In other words if the inequality is satisfied, then
P/Q ~ [ah <*i> • • ' t ön-l, An + c],
C ~ 0, ± 1,
where [#i, a2, • • • , ar, • • • ]=x is the infinite simple continued fraction for xf so that the at- are integers, a^ 1 ( i ^ 2 ) .
Suppose that x-p/q = e0/q2, 0 < 0 < 1, € = ± 1. Let
P/i = lbh *2, • ' * , bm],
p'/q' = [6i, b2, • • ' , bm-i],
where m (which we can choose to be odd or even) is taken so that
(— l) w ~ 1 = €. Defining y by the equation
oo = [ii, b2, • • • , bm y\ = (yp + p')/(yq + q')>
we obtain ^0 = q2(x—p/q) = (pfq—pq')q/(yq+q');
so that, since
p'q-pq' = (-l)™-i = e, y+q'/q = l/d.
Since 1 / 0 > 1 and q'/q<\ it follows that y>0.
If y>l,
then y=[&»+i, 6m+2, • • • ] fc+i^l, • • • ), and so
# = [6ii b2, - - • , &m, &m+i, • • • ], which, since the infinite simple continued fraction is unique, shows that p/q = [&i, • • • , bm] is the rath
convergent to x. If however y<\, then l/y = [c, 6 w+ i, bm+2f • • • ]
with £ ^ 1 . But <z/<z' = [i w , bm-u ' ' ' > b2] and therefore one of c and
bm must be unity for, if not, then l/y>2, q/q'>2,
y+q'/q<Kl/0.
1
Editor's note. In the meantime, R. M. Robinson has proved similar results in the
Duke Mathematical Journal, vol. 7 (1940), pp. 354-359. Also the first part of Theorem 1 was observed by P. Fatou, Comptes Rendus de l'Académie des Sciences, Paris,
vol. 139 (1904), pp. 1019-1021.
602
603
APPROXIMATIONS TO IRRATIONALS
Hence x= [fa, • • • , bm-i, bm+c, bm+i, • • • ] and &t- = az(i^m),
bm+c = am. Thus £ / g = [#i, a2, • • • , #m-i, &m] where &m = l or am — 1.
Consequently
^ / g = ( ^ m - l + ptn-2)/(qm-l
+ #m-2),
&m =
1,
#m = ^m ~
1.
or
#/<?
=
(#»» ~" Pm-l)/(qm
— #m-l),
The first part of the theorem is proved.
Now let p/q = (pn + epn-i)/(qn + eqn-i), c = ± 1 , Pn/qn being the nth.
convergent to x=[au • • • , an, x'] = (x'pn+pn-i)/(x'qn+qn-i).
Then
q2\ x - p/q\ = \ x'e - l\ (qn + egn_i)/\x''qn + gn_i),
which, since | e| = 1, x' > 1, is less than unity if and only if
€(>' -
qn/qn-i)
< 2.
But this inequality is certainly satisfied when e has the sign opposite
to the sign of x' — qn/qn-i. The second part of the theorem follows.
Irreducible fractions p/q can be divided into three classes [o/e],
[e/o], [o/o] in which o and e denote odd and even integers respectively.
Since pnQn-i—pn-iQn=i:l
it is clear that consecutive convergents pn-i/qn-i, pn/qn belong to two different classes and hence that
(pn + £pn-i)/(qn + eqn-i) where e = ± 1 must belong to the remaining
class of irreducible fractions. It follows from Theorem 1 that for
any irrational x infinitely many fractions of each class exist such that
\x—p/q\
<l/q2.
Theorem 1 in fact determines all such fractions.
This result is due to Scott 2 who used the geometric properties of
elliptic modular transformations. Scott also showed that the result
is the best possible: for a given class and a fixed k, 0 < & < 1 , irrationals exist, dense everywhere on the real axis, such that the inequality
\x — p/q\ <k/q2 is satisfied by only a finite number of fractions in the
given class.
To prove the last statement it will be enough to show that, if
x = \a\, #2, • • • , an, - ' * ] where the an are even integers not less than
2E + 1, where E>1, then for every fraction of type [o/o],
0 = q21 x - p/q \ > 1 -
1/E.
If 0 > 1 , there is nothing to prove. If 0 < 1 , it follows from our theo2
W. T. Scott, this Bulletin, vol. 46 (1940), pp. 124-129.
604
[August
M. F. SMILEY
rem that (p/q being irreducible)
P = pn + tpn-l,
q = qn+ €<7n-l,
€ = ± 1,
for the convergents to x are all [e/o] or [o/e]. Write X=[a w +i,
a n+2 , • • • ], F = [a», a n -i, • • • , a 2 ]. Then if n ^ 2 ,
(F + e)(X - c)
0=
2 - «(X - F)
=1
2+ X + F
> 1
,
XY + 1
ZF + 1
ZF + 1
X F + 1 - E(2 + X + Y) = (X - E)(Y - E) - £ 2 - 2E + 1
> (Jg + l) 2 - E 2 - 2£ + 1 > 0,
6 > 1-
1/E.
If » = 1, then /> = pi + l, 2 = ^1=1, 0 = 1 - [0, a2, • • • ] > l - l / £ .
R A F F L E S COLLEGE, SINGAPORE
MEASÜRABILITY AND DISTRIBUTIVITY IN THE
THEORY OF LATTICES 1
M. F . SMILEY
Introduction. Garrett Birkhoff2 derived the following self-dual symmetric condition that a metric lattice be distributive:
2\n(aUbKJ
c) - \x(aC\bC\c)\
= \t{a\JV) - v{aC\b)
+/x(aUc)
In a previous note 3 the author introduced and discussed a generalization of Carathéodory 's notion of measurability 4 with respect to an
outer measure function ju, which applies to arbitrary lattices L. The
ju-measurable elements form a subset L(JJ) consisting of those elements
# £ L which satisfy
(2)
p(a U 6) +
MO
r\ b) =
M(a)
+ /i(J)
for every 6 £ L . Closure properties of £(/x) were investigated. In par1
Presented to the Society, January 1, 1941. The author wishes to express his gratitude to the referee for his valuable suggestions and comments.
2
Lattice Theory, American Mathematical Society Colloquium Publications, vol.
25, p. 81. We shall adopt the notation and terminology of this work and shall indicate
specific references to it by B.
3
A note on measure f unctions in a lattice, this Bulletin, vol. 46 (1940), pp. 239-241.
We shall indicate references to this paper by M.
4
Vorlesungen Uber Réelle Funktionen, 2d edition, p. 246.